Existence and non-existence results of the Fucik type spectrum for the generalized $p$-Laplace operators (Progress in Variational Problems : Variational Methods in the Study of Evolution Equations)
Mieko Tanaka · Institutional Repositories DataBase (IRDB) · 2012
IntroductionIn this paper, we consider the existence of $(\alpha, \beta)\in \mathbb{R}^{2}$ for which the following quasilinear elliptic equation has a non-trivial solution: $(F)_{(\alpha,\beta)}$ $\{\begin{array}{ll}-divA(x, abla u)=\alpha u_{+}^{p-1}-\beta u_{-}^{p-1} in \Omega,\frac{\partial u}{\partial u}=0 on \partial\Omega,\end{array}$ where $ u$ denotes the outward unit normal vector on $\partial\Omega,$ $1 0$ for all $(x, t)\in\overline{\Omega}\cross(0, +\infty)$ and (i) $A\in C^{0}(\overline{\Omega}\cross \mathbb{R}^{N}, \mathbb{R}^{N})\cap C^{1}(\overline{\Omega}\cross(\mathbb{R}^{N}\backslash \{0\}), \mathbb{R}^{N})$ ; (ii) there exists a $C_{1}>0$ such that $|D_{y}A(x, y)|\leq C_{1}|y|^{p-2}$ for every $x\in\overline{\Omega}$ , and $y\in \mathbb{R}^{N}\backslash \{0\}$ ; (iii) there exists a $C_{0}>0$ such that $D_{y}A(x, y)\xi\cdot\xi\geq C_{0}|y|^{p-2}|\xi|^{2}$ for every $x\in\overline{\Omega},$ $y\in \mathbb{R}^{N}\backslash \{0\}$ and $\xi\in \mathbb{R}^{N}$ .